124 Professor Baker, On a property of focal conies 



bitangent circles of a second mode of generation, say ^ and S, 

 with points of contact on a circle, a. We shall prove that the 

 eight points of intersection of the pairs of circles (a, ^), (/S, y), 

 (y, §), (8, a) lie on two circles 6, 6', four on each. These circles 6, 6' 

 pass through the two intersections of the circles p, a, and separate 

 these circles harmonically; the circle p is orthogonal to the principal 

 circle to which the bitangent circles of the first mode are all ortho- 

 gonal, with a similar statement for cr. For the proof, let a = 0, 

 y = be the equations of any two tangent planes of a quadric 

 cone, whose generators of contact lie on a plane p = 0, so that 

 the cone has the equation ay — p^ = 0. Let ^S — o-^ = be another 

 quadric cone, whereof ^ = 0, 8 = are tangent planes touching 

 the cone on ct = 0. Then a quadric E = through the curve of 

 intersection of the two cones has an equation of the form 



E = ay -p^-m^ (^S - a^) = 0, 



so that the four lines a = 0, j3 = 0; ^ = 0, y=0; y = 0, 8 = 0; 

 8 = 0, a = 0, in which the two first planes a, y meet the two latter 

 planes ^, 8, intersect the quadric ^ = in eight points lying in 

 the two planes p + ma = 0, p — ma = 0. 



We have then a proof of Jessop's theorem in regard to the 

 bicircular quartic curve*. 



§ 3. Eeciprocally let any two conies be taken in space, not 

 intersecting one another. Consider a quadric touched by the 

 common tangent planes of these two conies. Then if A, C be any 

 two points of the first conic, and B, D any two points of the second 

 conic, it follows from § 2 that the pairs of tangent planes to this 

 quadric from the lines AB, BC, CD, DA touch two enveloping 

 cones of the quadric, say F and G. Or, as a line lying in a tangent 

 plane of a cone is a tangent line of the cone, there are two en- 

 veloping cones of the quadric which touch the lines AB, BC, CD, 

 DA. And, comparing the equations of § 2, the vertices of these 

 cones lie on the line joining the points R, S, in the planes of the 

 conies, which are the poles respectively of AC, BD in regard to 

 these conies, and separate R, S harmonically; the positions of the 

 vertices depend on the quadric taken to touch the common 

 tangent planes of the conies. Moreover, as the reciprocal of the 



* The direct analytical proof is, of course, simple. Let the fundamental quadric 

 be x^ + y~ + z^ + I" = 0, and bitangent circles of two modes be obtained by pro- 

 jection of the polar sections respectively of the two points 



[{a-d)x, (b-d)y, {c-d)z, 0], [{a-c)^, {b-c)v, 0, (c^-c)t]. 



Then the angle between these circles, being the Cayley separation of these points, 

 is the angle, in rectangular Cartesian coordinates, between the two lines 



X/px = Y/qy, X/p^ = Y/qr], where p^ = {a- d) [a - c), q^=(b- d) (b - c). 



This generalises at once to the Cyclide; cf. Jessop, Quartic surfaces, 1916, p. 106. 



